( Aside from its historic role as a total-computable-but-not-primitive-recursive function, Ackermann's original function is seen to extend the basic arithmetic operations beyond exponentiation, although not as seamlessly as do variants of Ackermann's function that are specifically designed for that purpose — such as Goodstein's hyperoperation sequence.

It is closely related to the Ackermann function and especially to the hyperoperation sequence.

Continuing in this manner leads to iterated exponentiation ( tetration ) and to the remainder of the hyperoperation sequence, which is commonly denoted using Knuth arrow notation.

He also introduced a variant of the Ackermann function that is now known as the hyperoperation sequence, together with the naming convention now used for these operations ( tetration, pentation, etc.

It is a hyperoperation and is non-commutative, and therefore has two inverse functions, which might be named the penta-root and the penta-logarithm ( analogous to the two inverse functions for exponentiation: nth root function and logarithm ).

The hyperstructures are algebraic structures equipped with at least one multi-valued operation, called a hyperoperation.

* Not displaying the password on the display screen as it is being entered or obscuring it as it is typed by using asterisks (*) or bullets (•).

(*) See commentary on Western League page.

Songs with an asterisk (*) are songs not composed by Barris, yet featured on the recordings:

Entries with a blue background and an asterisk (*) next to the writer's name have won the award, while use of a plus sign (+) instead indicates ties ; those with a white background are the other nominees on the shortlist.

for each p ∈ U. Any differential 1-form arises this way, and by using (*) it follows that any differential 1-form α on U may be expressed in coordinates as

(*) The Full Moon occurred on 28 January 1983 in UT, but on 29 January in Jerusalem mean local time ; however the full moon cycle and solar corrections are off from reality by about 3 hours, and put the syzygy back at 28 January in Jerusalem too.

* lyrics for the three (*) noted songs can be found on Dodd's official website under " The Battle of Vindaloo ", " A Crabby Lament " and " The song of Boot "

Like the dagger (†) and double dagger (‡), it is also sometimes used to link to a footnote where the asterisk (*) is already in use on a given page ; however, these usages are declining in favor of numbered footnotes, usually linked by a superscripted and / or square bracketed number.

Initially the British importations were identified on records with ( imp ) or an asterisk (*) added as a suffix to indicate that they were not locally bred.

Entries with a blue background and an asterisk (*) next to the writer's name have won the award ; those with a white background are the other nominees on the shortlist.

Entries with a blue background and an asterisk (*) next to the work's name have won the award ; those with a white background are the nominees on the short-list.

Entries with a blue background and an asterisk (*) next to the writer's name have won the award, while use of a plus sign (+) instead indicates ties ; those with a white background are the other nominees on the shortlist.

Entries with a blue background and an asterisk (*) next to the writer's name have won the award, while use of a plus sign (+) instead indicates ties ; those with a white background are the other nominees on the shortlist.

Entries with a blue background and an asterisk (*) next to the writer's name have won the award ; those with a white background are the other nominees on the shortlist.

Entries with a blue background and an asterisk (*) next to the writer's name have won the award ; those with a white background are the other nominees on the shortlist.

Entries with a blue background and an asterisk (*) next to the writer's name have won the award ; those with a white background are the other nominees on the shortlist.

Entries with a blue background and an asterisk (*) next to the artist's name have won the award ; those with a white background are the nominees on the short-list.

Entries with a blue background and an asterisk (*) next to the artist's name have won the award ; those with a white background are the nominees on the short-list.

Entries with a blue background and an asterisk (*) next to the writer's name have won the award ; those with a white background are the nominees on the short-list.

Entries with a blue background and an asterisk (*) next to the editor's name have won the award ; those with a white background are the nominees on the short-list.

NB: Names with (*) on the end were ' Messengers '.

* The (*) indicates animals only viewable on sky-ride

Villages marked with an asterisk (*) are now completely or partially on the Russian side:

Forgive me Gentlemen, if I talk a little extravagantly on these Matters ; I am no Divine: But if you, great Men, (*) must be making Laws, do not turn natural and useful Actions into Crimes, by your Prohibitions.

* Multiplication of two numbers ( computer languages bound to the rudimentary ASCII set often use an asterisk (*) to replace the missing crosses (×) and dots (·))

* Asterisks (*) of either a set length, or the length of the original word being filtered, like if " fuck " were to be filtered, it would appear as "****".

An asterisk (*) following a listing indicates a work done in partnership with H. Griffith Edwards.

This function is called the reproducing kernel for the Hilbert space H and it is determined entirely by H because the Riesz representation theorem guarantees, for every x in X, that the element K < sub > x </ sub > satisfying (*) is unique.

* top-level Usenet hierarchy, where the asterisk (*) is defined as a wildcard character.

For GOM input is free form with no sequence field and lines may be up to 255 characters long ; lines that start with an asterisk (*) are comments ; and lines that start with a plus-sign (+) are continuation lines.

The equation (*) is a result of the identity, for z

The remainder of this article describes the table-based kind of parser, the alternative being a recursive descent parser which is usually coded by hand ( although not always ; see e. g. ANTLR for an LL (*) recursive-descent parser generator ).

An LL parser is called an LL (*) parser if it is not restricted to a finite k tokens of lookahead, but can make parsing decisions by recognizing whether the following tokens belong to a regular language ( for example by use of a Deterministic Finite Automaton ).

It is easy to use (*) to confirm that (**) holds for x, y, z defined this way.

and if det ( A ) is a unit then (*) above shows that

This is indicated by an asterisk (*) in the list below.

(*) Note that the computed lunation of 29 26 / 49 days is a bit too long, so an adjustment of 1 / 49 day must be made periodically to compensate.

The asterisk (*) is generally used as a wildcard to replace a letter or letters.

(*) The word is not 無理 ( muri, " unfeasible "), which is normally used to indicate a situation that is locked, but possibly solvable, but 不可能 ( fukanou, " impossible ")-it's a much more definite term.

Typographical devices such as the asterisk (*) or dagger (†) may also be used to point to footnotes ; the traditional order of these symbols is *, †, ‡, §, ‖, ¶.

Then the final formula in (*) must be true, and φ is also true.

where (*) represents an excited state, and ΔE is the small energy difference between the energy states of the two atoms, of the order of 0. 05 eV or 387 cm < sup >− 1 </ sup >, which is supplied by kinetic energy.

A star (*) indicates that NCL is understood to have had significant control but not ownership: